UNITS, DIMENSION AND MEASUREMENT , MOTION …...Compendium UNITS, DIMENSION AND MEASUREMENT , MOTION...

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Transcript of UNITS, DIMENSION AND MEASUREMENT , MOTION …...Compendium UNITS, DIMENSION AND MEASUREMENT , MOTION...

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    UNITS, DIMENSION AND MEASUREMENT

    Measurement of large distance (Parallax Method)

    𝐷 = 𝑏

    𝜃

    Here

    D = distance of the planet from the earth.

    θ = parallax angle.

    b = distance between two place of observation

    Measurement of the size of a planet or a star.

    Here

    D = distance of planet from the earth,

    d = diameter of planet.

    α = angular diameter of planet.

    Measurement of mass

    The gravitational force on object, of mass m, is called the weight of the object.

    1 amu = 1.66 × 10-27 kg = 1u

    Estimation of Error

    Suppose the values obtained in several measurement of physical quantity

    are a1, a2, ............... an . Their arithmetic mean is

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    Absolute Error -

    Δa1, Δa2 ... , Δan called absolute error

    Average absolute error

    Fractional Error

    Percentage Error

    𝛿𝑎% =∆𝑎

    𝑎× 100%

    Combination of Errors

    ( i) Addition : Z = A + B

    Maximum Absolute Error = ΔA + Δ B

    Maximum Relative Error=

    ∆𝐴 + ∆𝐵

    𝐴 + 𝐵

    Maximum Percentage Error =

    (∆𝐴 + ∆𝐵

    𝐴 + 𝐵) × 100

    (ii) Subtractions : Z = A – B

    Maximum Absolute Error = ΔA + Δ B

    Maximum Relative Error = ∆𝐴+∆𝐵

    𝐴−𝐵

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    Maximum Percentage Error =

    (∆𝐴+∆𝐵

    𝐴−𝐵)×100

    (iii) Multiplication Z = A × B

    Maximum Absolute error = AΔB + BΔA

    Maximum Relative Error=

    ∆𝐴

    𝐴+

    ∆𝐵

    𝐵

    Maximum Percentage Error =

    (∆𝐴

    𝐴+

    ∆𝐵

    𝐵) × 100

    (iii) Division Z = A/B

    Maximum Absolute error =

    𝐵∆𝐴 + 𝐴∆𝐵

    𝐵2

    Maximum Relative Error =

    (iv) Power Z = An

    Maximum Absolute Error : n An-1ΔA

    Maximum Relative Error

    Maximum Percentage Error=

    𝑛∆𝐴

    𝐴× 100

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    Rule for determining number of significant figures

    (i) All the non - zero digits are significant

    (ii) All the zeros between two non zero digits are significant no matter where

    the decimal point is it at all.

    (iii) If the number is less then 1 then zeros on the right of decimal point but to

    the left of the first non - zero digit are not significant.

    (iv) In a number without decimal point the zeros on the right side of the last

    non zero digit are not significant.

    Dimensions and Dimensional formulas.

    (i) The expression of a physical quantity with appropriate powers of M, L, T, K,

    A etc is called the dimensional formula of that physical quantity.

    (ii) The power of exponents of M, L, T, K, A are called dimensions of that

    quantity.

    Some important units of distance

    1fermi (fm) =10–15m

    1Å = 10–10m

    1AU =1.496×1011m

    1light year = 9.46×1015m

    1par sec = 3.08×1016m

    Conversion of one System of units into another

    Units used : Every quantity must be expressed in its absolute units only

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    Principle of homogeneity of dimensions: Equating the power of M,L,T on

    either side of the equation OR power of dimension either side must be same

    Quantities having same dimensions/ units

    (i) Frequency, angular frequency, angular velocity, velocity gradient :

    M0 L0 T-1

    (ii) Work, Internal energy, P.E. , K.E., torque, moment of force : M1 L2 T-2

    (iii) Pressure, Stress, Young's modulus, Modulus of rigidity, Energy density :

    M1 L-1 T-2

    (iv) Mass and Inertia : M1 L0 T0

    (v) Momentum and Impulse : M1 L1 T-1

    (vi) Acceleration, g, gravitational intensity : M0 L1 T-2

    (vii) Thrust, Force, Weight, Energy radiant : M1 L1 T-2

    (viii) Angular momentum and Planck's constant (h) : M1 L2 T-1

    (ix) Surface tension, Surface energy ( energy per unit area), force gradient,

    Spring constant : M1L0T-2

    (x) Strain, Refractive index, relative density, angle, solid angle, distance

    gradient, relative permeability, relative permitivity,: M0 L2 T-2

    (xi) Thermal capacity, gas constant, Boltzmann constant and entropy :

    M1 L2 T-2 K-1

    (xii) L/R, √(LC) and RC : M0 L0 T1

    here R : resistance; C : capacitance; L : inductance

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    MOTION IN ONE DIMENSION

    Distance

    (i) It is the length of actual path traversed by a body during motion in a given

    interval of time

    (ii) Distance is a scalar quantity

    (iii) Value of distance travelled by moving body can never be zero or negative.

    Displacement

    (i) Displacement of a body in a given time is defined as the change in the

    position of the body in the particular direction during that time

    (ii) Displacement is vector quantity.

    (iii) The value of displacement can be zero or negative or positive

    (iv) The value of displacement can never be greater than the distance

    travelled.

    Displacement-time graph of various types of motion of a body (i) For

    stationary body

    Slope of straight line (representing instantaneous velocity) is zero.

    (ii) When body is moving with a constant velocity, or acceleration is zero

    straight line inclined to time axis

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    Greater is the slope of straight line higher is the velocity

    (iii) When body is moving with a constant positive acceleration, the time-

    displacement curve with bend upwards

    The slope of time-displacement curve increases with time or velocity is

    increasing with time

    (iv) When body is moving with a constant negative acceleration or constant

    retardation, the time-displacement curve with bend downwards

    The slope of time-displacement curve ( i.e. instantaneous velocity) decreases

    with time.

    (v) When a body is moving with infinite velocity, the time - displacement

    graph is parallel to displacement axis.

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    Such motion of a body is never possible

    (vi) When body returns back towards the original point of reference while

    moving with uniform negative velocity, the time displacement graph is an

    oblique straight line making angle θ > 90°

    Displacement of the body decreases with time with respect to the reference

    point, till it becomes zero

    Velocity - time graph o various types of motion of a body

    (i) When a body is moving with a constant velocity, the velocity - time graph is

    a straight line., parallel to time axis

    The slope of the graph , representing the instantaneous acceleration is zero

    (ii) When a body is moving with a constant acceleration and its initial velocity

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    is zero, the velocity time graph is an oblique straight line, passing through

    origin

    (iii) When a body is moving with constant speed with a constant acceleration

    and its initial velocity is not zero, the velocity - time graph is an oblique

    straight line not passing through origin

    Here OA represents the initial velocity of the body

    The area enclosed by the velocity-time graph with time axis represents the

    distance travelled by the body

    (iv) When a body is moving with a constant retardation and its initial velocity

    is not zero, the velocity - time graph is an oblique straight line, not passing

    through origin

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    The slope of the line is negative indicating retardation

    (v) When a body is moving with increasing acceleration , the velocity - time

    graph is a curve with bend upwards

    The slope of the graph increases with time. or instantaneous acceleration

    increases with time

    (vi) When a body is moving with decreasing acceleration, the velocity - time

    graph is a curve with bend downwards

    The slope of velocity - time graph decreases with time. or instantaneous

    acceleration decreases with time

    Acceleration - time graph of various types of motion of body

    (i) When a body is moving with constant acceleration, the acceleration - time

    graph is a straight line, parallel to time axis

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    The area enclosed by acceleration - time graph for the body for the given time

    gives change in velocity of the body for given time

    (ii) When a body is moving with constant increasing acceleration, the

    acceleration - time graph is straight line

    The body is moving with positive acceleration

    When a body is moving with constant decreasing acceleration , the

    acceleration - time graph is straight line

    The body is moving with negative acceleration and slope of straight line makes

    an angle θ > 90° with time axis. Or slope of the line is negative

    IMPORTANT FORMULAS

    (i)Equations of motion

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    u : initial velocity, v : final velocity , a : acceleration,

    t : time period, S: displacement

    𝑆 = 𝑢𝑡 +1

    2𝑎𝑡2

    𝑣 = 𝑢 + 𝑎𝑡

    𝑣2 = 𝑢2 + 2𝑎𝑠

    𝑆 = (𝑣 + 𝑢

    2) 𝑡

    𝑎 =𝑣 − 𝑢

    𝑡

    (ii) Free fall

    h: height, g = gravitational acceleration , final velocity :v, t: time period

    𝑣 = 𝑔𝑡

    𝑣2 = 2𝑔ℎ

    (iii) Object thrown up

    h: height, g = gravitational acceleration , u : initial velocity( negative) , t: time

    period

    𝑢 = √2𝑔ℎ

    ℎ =𝑢2

    2𝑔

    (iv) Distance travelled during last n sec while body is moving up = distance

    travelled during first n second of free fall

    (v) Distance travelled in nth second

    𝑆𝑛 = 𝑢 +𝑎

    2(2𝑛 − 1)

    (vi) If time period for two different section is same and v1 and v2 are the

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    velocities for two sections then average velocity is

    < 𝑉 > = 𝑉1 + 𝑉2

    2

    (vii) If an object covers distance 'x' with constant speed of v1 and same

    distance with constant speed of v2then average speed of is

    < 𝑉 > = 2𝑉1𝑉2𝑉1+𝑉2

    (viii) Starting from position of rest particle moves with constant accelerates +α

    reaches maximum velocity vmax then particle moves with constant decelerated

    β and become stationary . total time elapsed during this is t , then maximum

    velocity of particle is vmax is

    𝑉𝑚𝑎𝑥 = (𝛼𝛽

    𝛼 + 𝛽) 𝑡

    Calculus

    S: displacement , v : instantaneous velocity , a : acceleration

    𝑣 =𝑑𝑠

    𝑑𝑡

    𝑎 =𝑑𝑉

    𝑑𝑡

    if VA is magnitude of velocity of object A and VB is magnitude of velocity of

    object B with respect to stationary observer then

    (i) if both objects are moving in same direction, velocity of A with respect to B,

    VAB= VA - VB

    (ii) If both objects are approaching, velocity of A with respect to B ,

    VAB = VA + VB

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    MOTION IN TWO AND THREE DIMENSIONS

    Vector

    (i) If vector 𝐴 ⃗⃗ ⃗ = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧�̂� then magnitude of vector is

    (ii) If Vector 𝐴 ⃗⃗ ⃗ = 𝑥1𝑖̂ + 𝑦1𝑗̂ + 𝑧1�̂�

    and vector 𝐵 ⃗⃗ ⃗ = 𝑥2𝑖̂ + 𝑦2𝑗̂ + 𝑧2�̂�

    (a) are equal then

    x1 = x2

    y1 = y2

    z1 = z2

    (b) If �⃗� = 𝐴⃗⃗ ⃗ + �⃗� then

    𝑅 ⃗⃗ ⃗ = (𝑥1 + 𝑥2)𝑖̂ + (𝑦1 + 𝑦2)𝑗̂ + (𝑧1 + 𝑧2)�̂�

    (iii) If If 𝐴 ⃗⃗ ⃗ = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧�̂� is vector then

    (a) vector is making angle α with positive x-axis then

    𝑐𝑜𝑠𝛼 =𝑥

    √𝑥2 + 𝑦2 + 𝑧2=

    𝑥

    |𝐴 |

    (b) Vector is making angle β with positive y-axis then

    𝑐𝑜𝑠𝛽 =𝑦

    √𝑥2 + 𝑦2 + 𝑧2=

    𝑦

    |𝐴 |

    (c) Vector makes angle of γ with positive z-axis then

    𝑐𝑜𝑠𝛾 =𝑧

    √𝑥2 + 𝑦2 + 𝑧2=

    𝑧

    |𝐴 |

    (d) We can multiply and divide vector by scalar quantity but we can not divide

    vector by another vector

    (d) 𝑐𝑜𝑠2𝛼 + 𝑐𝑜𝑠2𝛽 + 𝑐𝑜𝑠2𝛾 = 1

    (iv) If 𝐴 ⃗⃗ ⃗ = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧�̂� is vector then unit vector n is

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